Intermath: Investigations: Algebra: Functions and Equations

Hansel has goldfish that quadruple, or become four times as many, every month. Gretel has goldfish thatincrease by 20 every month. Right now, Hansel has 4 goldfish and Gretel has 128 goldfish. In how many months, will they have the same number of goldfish? Show how you arrived at your answer.

So thinking of Hansel, he has four goldfish right now and next month he will have 4 x 4 goldfish. Or 16 goldfish. Then the next month he will have 16 x 4 goldfish, or 64 goldfish. A pattern is developing and it is important to find an equation that demonstrates our pattern.

4^n = goldfish

The equation is quite simple. The four represents the fact that the goldfish compound four times each month. The n is the number of the month from right now, assuming right now is month 1. So next month would be month 2, and so on. Therefore, when four is raised to the nth power, the answer will be the number of goldfish that have compounded over that number of months.

Now turning our attention to Gretel. She has 128 goldfish right now and will have 20 more each month there after. So on month 2 she would have 148, and month 3 would be 168. Again to represent this pattern, it is important to find an equation.

128 + 20n = goldfish

Beginning with the original 128 goldfish, that is added to 20n. The 20 is the number of fish added each month, and the n is the month (same as above). They are multiplied together because with each month there is an additional group of twenty goldfish that Gretel has.

The question asks after how many months will Hansel and Gretel have the same number of goldfish. The first equation shows how many goldfish Hansel will have with reference to each month, and the second equation shows how many goldfish Gretel will have in reference to each month. So they will have the same number of fish when the two equations are equal. If the two equations must be equal, then they can be set equal to each other. Now it is just a use of algebra:

4^n = 128 + 20n


There is a negative value for n as well, but that does not make sense in terms of our question.

So in answering the question, it would take 4 months for Hansel and Gretel to have the same number of goldfish.